synchronization of dissipative dynamical systems driven by non...

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2010, Article ID 502803, 13 pagesdoi:10.1155/2010/502803

Research ArticleSynchronization of Dissipative Dynamical SystemsDriven by Non-Gaussian Levy Noises

Xianming Liu,1 Jinqiao Duan,1, 2 Jicheng Liu,1and Peter E. Kloeden3

1 School of Mathematics and Statistics, Huazhong University of Science and Technology,Wuhan 430074, China

2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA3 Institut fur Mathematik, Johann Wolfgang Goethe-Universitat, D-60054, Frankfurt am Main, Germany

Correspondence should be addressed to Xianming Liu, [email protected]

Received 17 September 2009; Accepted 15 January 2010

Academic Editor: Salah-Eldin Mohammed

Copyright q 2010 Xianming Liu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Dynamical systems driven by Gaussian noises have been considered extensively in modeling,simulation, and theory. However, complex systems in engineering and science are often subjectto non-Gaussian fluctuations or uncertainties. A coupled dynamical system under a class of Levynoises is considered. After discussing cocycle property, stationary orbits, and random attractors, asynchronization phenomenon is shown to occur, when the drift terms of the coupled system satisfycertain dissipativity and integrability conditions. The synchronization result implies that coupleddynamical systems share a dynamical feature in some asymptotic sense.

1. Introduction

Synchronization of coupled dynamical systems is an ubiquitous phenomenon that has beenobserved in biology, physics, and other areas. It concerns coupled dynamical systems thatshare a dynamical feature in an asymptotic sense. A descriptive account of its diversityof occurrence can be found in the recent book [1]. Recently Caraballo and Kloeden [2, 3]proved that synchronization in coupled deterministic dissipative dynamical systems persistsin the presence of various Gaussian noises (in terms of Brownian motion), provided thatappropriate concepts of random attractors and stochastic stationary solutions are usedinstead of their deterministic counterparts.

2 International Journal of Stochastic Analysis

In this paper we investigate a synchronization phenomenon for coupled dynamicalsystems driven by nonGaussian noises. We show that couple dissipative systems exhibitsynchronization for a class of Levy motions.

This paper is organized as follows. We first recall some basic facts about randomdynamical systems (RDSs) as well as formulate the problem of synchronization of stochasticdynamical systems driven by Levy noises in Section 2. The main result (Theorem 3.3) and anexample are presented in Section 3.

Throughout this paper, the norm of a vector x in Euclidean space Rd is always denote

by |x|.

2. Dynamical Systems Driven by Levy Noises

Dynamical systems driven by nonGaussian Levy motions have attracted much attentionrecently [4, 5]. Under certain conditions, the SDEs driven by Levy motion generate stochasticflows [4, 6], and also generate random dynamical systems (or cocycles) in the sense of Arnold[7]. Recently, exit time estimates have been investigated by Imkeller and Pavlyukevich [8],and Imkeller et al. [9], and Yang and Duan [10] for SDEs driven by Levy motion. Thisshows some qualitatively different dynamical behavior between SDEs driven by Gaussianand nonGaussian noises.

2.1. Levy Processes

A Levy process or motion on Rd is characterized by a drift parameter γ ∈ R

d, a covarianced × d matrix A, and a nonnegative Borel measure ν, defined on (Rd,B(Rd)) and concentratedon R

d \ {0}, which satisfies

∫Rd\{0}

(∣∣y∣∣2 ∧ 1)ν(dy

)< ∞, (2.1)

or equivalently

∫Rd\{0}

∣∣y∣∣21 +

∣∣y∣∣2 ν(dy

)< ∞. (2.2)

This measure ν is the so-called the Levy jumpmeasure of the Levy process Lt. Moreover Levyprocess Lt has the following Levy-Ito decomposition:

Lt = γt + Bt +∫|x|<1

xN(t, dx) +∫|x|≥1

xN(t, dx), (2.3)

where N(dt, dx) is Poisson random measure,

N(dt, dx) = N(dt, dx) − ν(dx)dt, (2.4)

International Journal of Stochastic Analysis 3

is the compensated Poisson random measure of Lt, and Bt is an independent Brownianmotion on R

d with covariance matrixA (see [4, 10–12]). We call (A, ν, γ) the generating triplet.General semimartingales, especially Levy motions, are thought to be appropriate

models for nonGaussian processes with jumps [11]. Let us recall that a Levy motion Lt isa nonGaussian process with independent and stationary increments. Moreover, its samplepaths are only continuous in probability, namely, P(|Lt − Lt0 | ≥ δ) → 0 as t → t0 forany positive δ. With a suitable modification [4], these paths may be taken as cadlag, thatis, paths are continuous on the right and have limits on the left. This continuity is weakerthan the usual continuity in time. In fact, a cadlag function has finite or at most countablediscontinuities on any time interval (see, e.g., [4, page 118]). This generalizes the Brownianmotion Bt, since Bt satisfies all these three conditions, but additionally, (i) almost every samplepath of the Brownian motion is continuous in time in the usual sense, and (ii) the incrementsof Brownian motion are Gaussian distributed.

The next useful lemma provides some important pathwise properties of Lt with two-sided time t ∈ R.

Lemma 2.1 (pathwise boundedness and convergence). Let Lt be a two-sided Levy motion on Rd

for which E|L1| < ∞ and EL1 = 0. Then we have the following.

(i) limt→±∞(1/t)Lt = 0, a.s.

(ii) The integrals∫ t−∞e

−λ(t−s) dLs(ω) are pathwisely uniformly bounded in λ > 1 on finite timeintervals [T1, T2] in R.

(iii) The integrals∫ tT1e−λ(t−s) dLs(ω) → 0 as λ → ∞, pathwise on finite time intervals [T1, T2]

in R.

Proof. (i) This convergence result comes from the law of large numbers, in [11, Theorem36.5].

(ii) Since the function h(t) = e−λt is continuous in t, integrating by parts we obtain

∫ t

−∞e−λ(t−s) dLs(ω) = Lt(ω) − λ

∫ t

−∞e−λ(t−s)Ls(ω)ds. (2.5)

Then by (i) and the fact that every cadlag function is bounded on finite closed intervals, weconclude (ii).

(iii) Integrating again by parts, it follows that

∫ t

T1

e−λ(t−s) dLs(ω) = (Lt − LT1)e−λ(t−T1) + λ

∫ t

T1

e−λ(t−s)(Lt(ω) − Ls(ω))ds, (2.6)

from which the result follows.

Remark 2.2. The assumptions on Lt in the above lemma are satisfied by a wide class of Levyprocesses, for instance, the symmetric α-stable Levy motion on R

d with 1 < α < 2. Indeed, inthis case, we have

∫|x|> 1|x|ν(dx) < ∞, and then E|L1| < ∞, see [11, Theorem 25.3].

For the canonical sample space of Levy processes, that is, Ω = D(R,Rd) of cadlagfunctions which are defined on R and taking values in R

d is not separable, if we use


the usual compact-open metric. However, it is complete and separable when endowed withthe Skorohod metric (see, e.g., [13], [14, page 405]), in which case we call D(R,Rd) aSkorohod space.

2.2. Random Dynamical Systems

Following Arnold [7], a random dynamical system (RDS) on a probability space (Ω,F,P)consists of two ingredients: a driving flow θt on the probability space Ω, that is, θt is adeterministic dynamical system, and a cocycle mapping ϕ : R × Ω × R

d → Rd, namely, ϕ

satisfies the conditions

ϕ(0, ω) = idRd , ϕ(t + s,ω) = ϕ(t, θsω) ◦ ϕ(s,ω) (2.7)

for allω ∈ Ω and all s, t ∈ R. This cocycle is required to be at least measurable from the σ-fieldB(R) × F × B(Rd) to the σ-field B(Rd).

For random dynamical systems driven by Levy noise we take Ω = D(R,Rd) withthe Skorohod metric as the canonical sample space and denote by F := B(D(R,Rd)) theassociated Borel σ-field. Let μL be the (Levy) probability measure on F which is given bythe distribution of a two-sided Levy process with paths in D(R,Rd).

The driving system θ = (θt, t ∈ R) on Ω is defined by the shift

(θtω)(s) := ω(t + s) −ω(t). (2.8)

The map (t, ω) → θtω is continuous, thus measurable ([7, page 545]), and the (Levy)probability measure is θ-invariant, that is,

μL

(θ−1t (A)

)= μL(A) (2.9)

for all A ∈ F, see [4, page 325].We say that a family A = {A(ω), ω ∈ Ω} of nonempty measurable compact subsets

A(ω) of Rd is invariant for a RDS (θ, ϕ), if ϕ(t, ω,A(ω)) = A(θtω) for all t > 0 and that it is a

random attractor if in addition it is pathwise pullback attracting in the sense that

H∗d

(ϕ(t, θ−tω,D(θ−tω)), A(ω)

) −→ 0 as t −→ ∞ (2.10)

for all suitable families (called the attracting universe) of D = {D(ω), ω ∈ Ω} of nonemptymeasurable bounded subsets D(ω) of R

d, where H∗d(A,B) = supx∈Ainfy∈B|x − y| is the

Hausdorff semi-distance on Rd.

The following result about the existence of a random attractor may be proved similarlyas in [2, 15–18].


Lemma 2.3 (random attractor for cadlag RDS). Let (θ, ϕ) be an RDS on Ω × Rd and let ϕ be

continuous in space, but cadlag in time. If there exits a family B = {B(ω), ω ∈ Ω} of nonemptymeasurable compact subsets B(ω) of R

d and a TD,ω ≥ 0 such that

ϕ(t, θ−tω,D(θ−tω)) ⊂ B(ω), ∀t ≥ TD,ω, (2.11)

for all families D = {D(ω), ω ∈ Ω} in a given attracting universe, then the RDS (θ, ϕ) has a randomattractor A = {A(ω), ω ∈ Ω} with the component subsets defined for each ω ∈ Ω by

A(ω) =⋂s>0

⋃t≥s

ϕ(t, θ−tω, B(θ−tω)). (2.12)

Forevermore if the random attractor consists of singleton sets, that is, A(ω) = {X∗(ω)} for somerandom variable X∗, then X∗

t (ω) = X∗(θtω) is a stationary stochastic process.

We also need the following Gronwall’s lemma from [19].

Lemma 2.4. Let x(t) satisfy the differential inequality

d

dt+x ≤ g(t)x + h(t), (2.13)

where (d/dt+)x := limh↓0+((x(t + h) − x(t))/h) is right-hand derivative of x. Then

x(t) ≤ x(0) exp

[∫ t

0g(r)dr

]+∫ t

0exp

[∫ t

s

g(r)dr

]h(s)ds. (2.14)

2.3. Dissipative Synchronization

Suppose that we have two autonomous ordinary differential equations in Rd,

dx

dt= f(x),

dy

dt= g

(y), (2.15)

where the vector fields f and g are sufficiently regular (e.g., differentiable) to ensure theexistence and uniqueness of local solutions, and additionally satisfy one-sided dissipativeLipschitz conditions

max{⟨x1 − x2, f(x1) − f(x2)

⟩,⟨x1 − x2, g(x1) − g(x2)

⟩} ≤ −l|x1 − x2|2 (2.16)

on Rd for some l > 0. These dissipative Lipschitz conditions ensure existence and uniqueness

of global solutions. Each of the systems has a unique globally asymptotically stable equilibria,x and y, respectively [18]. Then, the coupled deterministic dynamical system in R

2d

dx

dt= f(x) + λ

(y − x

),

dy

dt= g(x) + λ

(x − y

)(2.17)


with parameter λ > 0 also satisfies a one-sided dissipative Lipschitz condition and, hence,also has a unique equilibrium (xλ, yλ), which is globally asymptotically stable [18]. Moreover,(xλ, yλ) → (z, z) as λ → ∞, where z is the unique globally asymptotically stable equilibriumof the “averaged” system in R

d

dz

dt=

12(f(z) + g(z)

). (2.18)

This phenomenon is known as synchronization for the coupled deterministic system (2.17).The parameter λ often appears naturally in the context of the problem under consideration.For example in control theory it is a control parameter which can be chosen by the engineer,whereas in chemical reactions in thin layers separated by a membrane it is the reciprocal ofthe thickness of the layers; see [20].

Caraballo and Kloeden [2], and Caraballo et al. [3] showed that this synchronizationphenomenon persists under Gaussian Brownian noise, provided that asymptotically stablestochastic stationary solutions are considered rather than asymptotically stable steady statesolutions. Recall that a stationary solution X∗ of a SDE system may be characterized as astationary orbit of the corresponding random dynamical system (θ, ϕ) (defined by the SDEsystem), namely, ϕ(t, ω,X∗(ω)) = X∗(θtω).

The aim of this paper is to investigate synchronization under nonGaussian Levy noise.In particular, we consider a coupled SDE system in R

d, driven by Levy motion

dXt =(f(Xt) + λ(Yt −Xt)

)dt + adL1

t ,

dYt =(g(Yt) + λ(Xt − Yt)

)dt + bdL2

t ,(2.19)

where a, b ∈ Rd are constant vectors with no components equal to zero, L1

t , L2t are

independent two-sided scalar Levy motion as in Lemma 2.1, and f, g satisfy the one-sideddissipative Lipschitz conditions (2.16).

In addition to the one-sided Lipschitz dissipative condition (2.16) on the functions fand g, as in [2] we further assume the following integrability condition. There exists m0 > 0such that for any m ∈ (0, m0], and any cadlag function u : R → R

d with subexponentialgrowth it follows

∫ t

−∞ems

∣∣f(u(s))∣∣2ds < ∞,

∫ t

−∞ems

∣∣g(u(s))∣∣2ds < ∞. (2.20)

Without loss of generality, we assume that the one-sided dissipative Lipschitz constant l ≤ m0.In the next section we will show that the coupled system (2.19) has a unique stationary

solution (Xλt , Y

λt )which is pathwise globally asymptotically stable with (Xλ

t , Yλt ) → (Z∞

t , Z∞t )

as λ → ∞, pathwise on finite time intervals [T1, T2], where Z∞t is the unique pathwise

globally asymptotically stable stationary solution of the “averaged” SDE in Rd

dZt =12[f(Zt) + g(Zt)

]dt +

12adL1

t +12bdL2

t . (2.21)


3. Systems Driven by Levy Noise

For the coupled system (2.19), we have the following two lemmas about its stationarysolutions.

Lemma 3.1 (existence of stationary solutions). If the Assumption (2.20) holds, f and g arecontinuous and satisfy the one-sided Lipschitz dissipative conditions (2.16) with Lipschitz constant l,then the coupled stochastic system (2.19) has a unique stationary solution.

Proof. First, the stationary solutions of the Langevin equations [4, 21]

dXt = −λXtdt + adL1t , dYt = −λYtdt + bdL2

t (3.1)

are given by

Xλ

t = ae−λt∫ t

−∞eλsdL1

t , Yλ

t = be−λt∫ t

−∞eλsdL2

t . (3.2)

The differences of the solutions of (2.19) and these stationary solutions satisfy a system ofrandom ordinary differential equations, with right-hand derivative in time

d

dt+

(Xt −X

λ

t

)= f(Xt) + λ(Yt −Xt) + λX

λ

t ,

d

dt+

(Yt − Y

λ

t

)= g(Yt) + λ(Xt − Yt) + λY

λ

t .(3.3)

The equations (3.3) are equivalent to

d

dt+Uλ

t = f(Xt) + λ(V λt −Uλ

t

)+ λY

λ

t ,d

dt+V λt = g(Yt) + λ

(Uλ

t − V λt

)+ λX

λ

t , (3.4)

where Uλt = Xt −X

λ

t and V λt = Yt − Y

λ

t . Thus,

12d

dt+

(∣∣∣Uλt

∣∣∣2 +∣∣∣V λ

t

∣∣∣2)

=(Uλ

t , f

(Uλ

t +Xλ

t

)− f

(X

λ

t

))+(V λt , g

(V λt + Y

λ

t

)− g

(Y

λ

t

))

+(Uλ

t , f

(X

λ

t

)+ λY

λ

t

)+(V λt , g

(Y

λ

t

)+ λX

λ

t

)− λ

∣∣∣Uλt − V λ

t

∣∣∣2

≤ − l

2

(∣∣∣Uλt

∣∣∣2 +∣∣∣V λ

t

∣∣∣2)+2l

∣∣∣∣f(X

λ

t

)+ λY

λ

t

∣∣∣∣2

+2l

∣∣∣∣g(Y

λ

t

)+ λX

λ

t

∣∣∣∣2

.

(3.5)


Hence, by Lemma 2.4,

∣∣∣Uλt

∣∣∣2 +∣∣∣V λ

t

∣∣∣2 ≤(∣∣∣Uλ

t0

∣∣∣2 +∣∣∣V λ

t0

∣∣∣2)el(t−t0)

+4e−lt

l

∫ t

t0

els[∣∣∣∣f

(X

λ

t

)+ λY

λ

t

∣∣∣∣2

+∣∣∣∣g(Y

λ

t

)+ λX

λ

t

∣∣∣∣2]ds.

(3.6)

Define

|Rλ(ω)|2 = 1 +4l

∫0

−∞els

[∣∣∣∣f(X

λ(θsω)

)+ λY

λ(θsω)

∣∣∣∣2

+∣∣∣∣g(Y

λ(θsω)

)+ λX

λ(θsω)

∣∣∣∣2]ds

(3.7)

and let Bλ2d(ω) be a random closed ball in R

2d centered on the origin and of radius Rλ(ω).Now we can use pathwise pullback convergence (i.e., with t0 → −∞) to show that

|Uλt |

2+ |V λ

t |2is pathwise absorbed by the family Bλ

2d = {Bλ2d(ω), ω ∈ Ω}, that is, for appropriate

families D, there exists TD,ω ≥ 0 such that

ϕ(t, θ−tω,D(θ−tω)) ⊂ Bλ2d(ω), ∀t ≥ TD,ω. (3.8)

Hence, by Lemma 2.3, the coupled system has a random attractor Aλ = {Aλ(ω), ω ∈Ω}with Aλ(ω) ⊂ Bλ

2d(ω).Note that, by Lemma 2.1, it can be shown that the random compact absorbing balls

Bλ2d(ω) are contained in the common compact ball for λ ≥ 1.

However, the difference (ΔXt,ΔYt) = (X1t −X2

t , Y1t −Y 2

t ) of any pair of solutions satisfiesthe system of random ordinary differential equations

d

dt+ΔXt = f

(X1

t

)− f

(X2

t

)+ λ(ΔYt −ΔXt),

d

dt+ΔYt = g

(Y 1t

)− g

(Y 2t

)− λ(ΔYt −ΔXt),

(3.9)

so

d

dt+

(|ΔXt|2 + |ΔYt|2

)= 2

(ΔXt, f

(X1

t

)− f

(X2

t

))+ 2

(ΔYt, g

(Y 1t

)− g

(Y 2t

))

− 2λ|ΔXt −ΔYt|2

≤ −2l(|ΔXt|2 + |ΔYt|2

)(3.10)

from which we obtain

|ΔXt|2 + |ΔYt|2 ≤(|ΔX0|2 + |ΔY0|2

)e−2lt (3.11)


which means all solutions converge pathwise to each other as t → ∞. Thus the randomattractor consists of a singleton set formed by an ordered pair of stationary processes(Xλ

t (ω), Y λt (ω)) or equivalently (Xλ(θtω), Y λ(θtω)).

Lemma 3.2 (a property of stationary solutions). The stationary solutions of the coupled stochasticsystem (2.19) have the following asymptotic behavior:

Xλt (ω) − Y λ

t (ω) −→ 0 as λ −→ ∞ (3.12)

pathwise on any bounded time interval [T1, T2] of R.

Proof. Since

d(Xλ

t − Y λt

)=(−2λ

(Xλ

t − Y λt

)+ f

(Xλ

t

)− g

(Y λt

))dt + adL1

t − bdL2t , (3.13)

we have

d(Dλ

t e2λt

)= e2λt

(f(Xλ

t

)− g

(Y λt

))+ ae2λtdL1

t − be2λtdL2t , (3.14)

where Dλt = Xλ

t − Y λt , so pathwise

∣∣∣Dλt

∣∣∣ ≤ e−2λ(t−T1)∣∣∣Dλ

T1

∣∣∣ +∫ t

T1

e−2λ(t−s)(∣∣∣f(Xλ

s

)∣∣∣ + ∣∣∣g(Y λs

)∣∣∣)ds

+ |a|∣∣∣∣∣∫ t

T1

e−2λ(t−s)dL1t

∣∣∣∣∣ + |b|∣∣∣∣∣∫ t

T1

e−2λ(t−s)dL2t

∣∣∣∣∣.(3.15)

By Lemma 2.1 we see that the radius Rλ(θtω) is pathwise uniformly bounded on eachbounded time interval [T1, T2], so we see that the right hand of above inequality convergeto 0 as λ → ∞ pathwise on the bounded time interval [T1, T2].

We now present the main result of this paper.

Theorem 3.3 (synchronization under Levy noise). Suppose that the coupled stochastic system inR

2d

dXt =(f(Xt) + λ(Yt −Xt)

)dt + adL1

t ,

dYt =(g(Yt) + λ(Xt − Yt)

)dt + bdL2

t

(3.16)

defines a random dynamical system (θ, ϕ). In addition, assume that the continuous functions f, gsatisfy the integrability condition (2.20) as well as the one-sided Lipschitz dissipative condition (2.16),then the coupled stochastic system (3.16) is synchronized to a single averaged SDE in R

d

dZt =12[f(Zt) + g(Zt)

]dt +

a

2dL1

t +b

2dL2

t , (3.17)


in the sense that the stationary solutions of (3.16) pathwise converge to that of (3.17), that is,(Xλ

t , Yλt ) → (Z∞

t , Z∞t ) pathwise on any bounded time interval [T1, T2] as parameter λ → ∞.

Proof. It is enough to demonstrate the result for any sequence λn → ∞. Define

Zλt :=

12

[Xλ

t + Y λt

], t ∈ R. (3.18)

Note that Zλt (ω) = Zλ(θtω) satisfies the equation

dZλt =

12

[f(Xλ

t

)+ g

(Y λt

)]dt +

a

2dL1

t +b

2dL2

t . (3.19)

Also we define

Zt(ω) = Z(θtω) :=12

[Xt(ω) + Yt(ω)

], t ∈ R, (3.20)

where Xt and Yt are the (stationary) solutions of the Langevin equations

dXt = −Xtdt + adL1t , dYt = −Ytdt + bdL2

t , (3.21)

that is,

Xt = ae−t∫ t

−∞esdL1

t , Y t = be−t∫ t

−∞esdL2

t . (3.22)

The difference Zλt − Zt satisfies

2(Zλ

t − Zt

)= 2

(Zλ − Z

)+∫ t

0

(f(Xλ

s

)+ g

(Y λs

)+Xs + Ys

)ds. (3.23)

By Lemma 2.1, and the fact that these solutions belong to the common compact ball and everycadlag function is bounded on finite closed intervals, we obtain

∣∣∣f(Xλt (ω)

)+ g

(Y λt (ω)

)∣∣∣ +∣∣∣Xt(ω) + Yt(ω)

∣∣∣ ≤ MT1,T2(ω) < ∞, (3.24)

which implies uniform boundedness as well as equicontinuity. Thus by the Ascoli-Arzelatheorem [13], we conclude that for any sequence λn → ∞, there is a random subsequence


λnj (ω) → ∞, such thatZλnjt (ω)−Zt(ω) → Z∞

t (ω)−Zt(ω) as j → ∞. ThusZλnjt (ω) → Z∞

t (ω)as j → ∞. Now, by Lemma 3.2

Zλnjt (ω) − Y

λnjt (ω) =

Xλnjt (ω) − Y

λnjt (ω)

2−→ 0,

Zλnjt (ω) − X

λnjt (ω) =

Yλnjt (ω) − X

λnjt (ω)

2−→ 0,

(3.25)

as λnj → ∞, so

Xλnjt (ω) = 2Z

λnjt (ω) − Y

λnjt (ω) −→ Z∞

t (ω),

Yλnjt (ω) = 2Z

λnjt (ω) − X

λnjt (ω) −→ Z∞

t (ω),(3.26)

as λnj → ∞.Using the integral representation of the equation, it can be verified thatZ∞

t is a solutionof the averaged random differential equation (3.17) for all t ∈ R. The drift of this SDE satisfiesthe dissipative one-sided condition (2.16). It has a random attractor consisting of a singletonset formed by a stationary orbit, which must be equal to Z∞

t .Finally, we note that all possible subsequences of Zλn

t have the same pathwise limit.Thus the full sequence Zλn

t converges to Z∞t , as λn → ∞. This completes the proof.

3.1. An Example

Consider two scalar SDEs:

dXt = −(Xt + 1)dt + dL1t , dYt = −(Yt + 3)dt + 2dL2

t , (3.27)

which we rewrite as

dXt = −Xtdt + dL3t , dYt = −Ytdt + 2dL4

t , (3.28)

where L3t = −t + L1

t and L4t = −3t/2 + L2

t .The corresponding coupled system (3.16) is

dXt = −Xtdt + λ(Yt −Xt)dt + dL3t ,

dYt = −Ytdt + λ(Xt − Yt)dt + 2dL4t

(3.29)


with the stationary solutions

Xλt =

∫ t

−∞e−(λ+1)(t−s) coshλ(t − s)dL3

s + 2∫ t

−∞e−(λ+1)(t−s) sinhλ(t − s)dL4

s,

Y λt =

∫ t

−∞e−(λ+1)(t−s) sinhλ(t − s)dL3

s + 2∫ t

−∞e−(λ+1)(t−s) coshλ(t − s)dL4

s.

(3.30)

Let λ → ∞, then

(Xλ

t , Yλt

)−→ (Z∞

t , Z∞t ), (3.31)

where Z∞t , given by

Z∞t =

∫ t

−∞

12e−(t−s)dL3

s +∫ t

−∞e−(t−s)dL4

s, (3.32)

is the stationary solution of the following averaged SDE:

dZt = −Ztdt +12dL3

t + dL4t , (3.33)

which is equivalent to the following SDE, in terms of the original Levy motions L1 and L2,

dZt = −(Zt + 2)dt +12dL1

t + dL2t . (3.34)

Acknowledgments

The authors would like to thank Peter Imkeller and Bjorn Schmalfuss for helpful discussionsand comments. This research was partly supported by the NSF Grants 0620539 and 0731201,NSFC Grant 10971225, and the Cheung Kong Scholars Program.

References

[1] S. Strogatz, Sync: The Emerging Science of Spontaneous Order, Hyperion Books, New York, NY, USA,2003.

[2] T. Caraballo and P. E. Kloeden, “The persistence of synchronization under environmental noise,”Proceedings of The Royal Society of London. Series A, vol. 461, no. 2059, pp. 2257–2267, 2005.

[3] T. Caraballo, P. E. Kloeden, and A. Neuenkirch, “Synchronization of systems with multiplicativenoise,” Stochastics and Dynamics, vol. 8, no. 1, pp. 139–154, 2008.

[4] D. Applebaum, Levy Processes and Stochastic Calculus, vol. 93 of Cambridge Studies in AdvancedMathematics, Cambridge University Press, Cambridge, UK, 2004.

[5] D. Schertzer, M. Larcheveque, J. Duan, V. V. Yanovsky, and S. Lovejoy, “Fractional Fokker-Planckequation for nonlinear stochastic differential equations driven by non-Gaussian Levy stable noises,”Journal of Mathematical Physics, vol. 42, no. 1, pp. 200–212, 2001.


[6] H. Kunita, “Stochastic differential equations based on Levy processes and stochastic flows ofdiffeomorphisms,” in Real and Stochastic Analysis, Trends Math., pp. 305–373, Birkhauser, Boston,Mass, USA, 2004.

[7] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer, Berlin,Germany, 1998.

[8] P. Imkeller and I. Pavlyukevich, “First exit times of SDEs driven by stable Levy processes,” StochasticProcesses and Their Applications, vol. 116, no. 4, pp. 611–642, 2006.

[9] P. Imkeller, I. Pavlyukevich, and T. Wetzel, “First exit times for Levy-driven diffusions withexponentially light jumps,” The Annals of Probability, vol. 37, no. 2, pp. 530–564, 2009.

[10] Z. Yang and J. Duan, “An intermediate regime for exit phenomena driven by non-Gaussian Levynoises,” Stochastics and Dynamics, vol. 8, no. 3, pp. 583–591, 2008.

[11] K.-I. Sato, Levy Processes and Infinitely Divisible Distributions, vol. 68 of Cambridge Studies in AdvancedMathematics, Cambridge University Press, Cambridge, UK, 1999.

[12] G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models withInfinite Variance, Stochastic Modeling, Chapman & Hall, New York, NY, USA, 1994.

[13] P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York, NY, USA, 1968.[14] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Mathematical and

Analytical Techniques with Applications to Engineering, Springer, New York, NY, USA, 2005.[15] B. Schmalfuss, “Attractors for nonautonomous and random dynamical systems perturbed by

impulses,” Discrete and Continuous Dynamical Systems. Series A, vol. 9, no. 3, pp. 727–744, 2003.[16] H. Crauel and F. Flandoli, “Attractors for random dynamical systems,” Probability Theory and Related

Fields, vol. 100, no. 3, pp. 365–393, 1994.[17] B. Schmalfuß, “The random attractor of the stochastic Lorenz system,” Zeitschrift fur Angewandte

Mathematik und Physik, vol. 48, no. 6, pp. 951–975, 1997.[18] P. E. Kloeden, “Nonautonomous attractors of switching systems,” Dynamical Systems, vol. 21, no. 2,

pp. 209–230, 2006.[19] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and

the Theory of Global Attractor, Cambridge Texts in Applied Mathematics, Cambridge University Press,Cambridge, UK, 2001.

[20] T. Caraballo, I. D. Chueshov, and P. E. Kloeden, “Synchronization of a stochastic reaction-diffusionsystem on a thin two-layer domain,” SIAM Journal on Mathematical Analysis, vol. 38, no. 5, pp. 1489–1507, 2007.

[21] O. E. Barndorff-Nielsen, J. L. Jensen, and M. Sørensen, “Some stationary processes in discrete andcontinuous time,” Advances in Applied Probability, vol. 30, no. 4, pp. 989–1007, 1998.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

synchronization of dissipative dynamical systems driven by non...

Documents